04-07-2014, 02:24 PM
Scaling Behaviour of Pressure-Driven Micro-Hydraulic Systems
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ABSTRACT
This paper presents a lumped network approach for the
modelling and design of micro-hydraulic systems. A
hydraulic oscillator has been built consisting of hydraulic
resistors, capacitors and transistors (pressure controlled
valves). The scaling of micro-hydraulic networks consisting
of linear resistors, capacitors and inertances has been
studied. An important result is that to make smaller
networks faster, driving pressures should increase with
reducing size.
INTRODUCTION
Micro-hydraulic systems can be modeled and designed
using a generalized physical system description [1, 2]. This
approach is based on the assumption that it is possible to
separate and concentrate properties of a system into
interconnected subsystems. It has proven its great value in
the design of electronic circuits. The lumped network
approach also offers a powerful design tool for microfluidic
systems [3-5]. To illustrate the far-reaching analogy
between different physical domains, we have rebuilt an
electronic astable multivibrator network in the hydraulic
domain [4]. The system consists of hydraulic capacitors,
resistors, transistors and (parasitic) coils. Based on this
micro-hydraulic system the scaling behaviour of low Re
(Reynolds number) hydraulic systems has been analyzed
2 HYDRAULIC FUNDAMENTALS
In every physical domain a conserved quantity q can be
distinguished [1]. The flow is the rate of exchange of this
conserved quantity between subsystems. In the hydraulic
domain the flow variable is the volume flow φV [m3s-1]. The
effort is the tension that governs the exchange of the
conserved quantity between subsystems. In the hydraulic
domain the effort variable is the pressure p [Pa].
Hydraulic Resistors
The hydraulic resistor physically is a liquid flow
restriction, symbolically represented as in fig. 1a. For a
linear flow resistor, the resistance R is defined by:
HYDRAULIC RELAXATION OSCILLATOR
Fig. 4 shows the schematic of a hydraulic astable
multivibrator. It consists of a series connection of two
inverting amplifiers and two relaxation coupling networks .
The two inverting amplifiers are represented by PCV1, R5
and by PCV2, R6. The two RC relaxation networks are
represented by R1, R2, C1 and R3, R4, C2. Hydraulic
'transistors' are implemented by means of a pressure
controlled membrane valve (PCV). In the schematic shown
in fig. 4 the transistors PCV1 and PCV2 should have a low
resistance (open state) when the control pressure is low, in
analogy with p-MOS (Metal-Oxide-Semiconductor)
transistors
3.2 Oscillator System Behaviour
All components from the schematic fig. 4 were realized in
glass-silicon-glass technology on a single wafer, and
connected by plastic tubes (fig. 7). The frequency of
oscillation is determined by the relaxation time-constants
R1//R2⋅C1 and R3//R4⋅C2. The values chosen for the
resistors are listed in table 1, including the size of the
triangular resistor channels. For the capacitors C1,2 a value
of 1 x 10-12 N-1m5 was chosen, which corresponds with a
silicon membrane radius of 6.5 mm in combination with a
42 μm thickness. Together with the resistor values chosen a
oscillation frequency of 0.16 Hz was predicted by network
simulation [4]. The multivibrator was tested successfully,
driven by a constant ethanol pressure of 0.1 bar (fig. 8). The
measured free-running oscillation frequency was 0.18 Hz,
in good agreement with the prediction.
Down-Scaling with Increasing Pressure
Smaller R⋅C time-constants can be obtained by
increasing the stiffness of capacitor membranes. We
therefore propose to down-scale all dimensions λ times,
except the membrane thickness. If the membrane thickness
scales down with λ2/3 we find C' = C / λ5 and therefore
(R⋅C)' = (R⋅C) / λ2. Because V' = V / λ3, it follows from eq.
(5) that the pressure scales with p' = pλ2
. Again for the
inertance related relaxation we find (I/R)'=(I/R)/λ2. Both
time-constants scale down with λ2
, which implies that if
pressures increase with decreasing size hydraulic systems
indeed can be made quicker. For liquids there is the
practical limit caused by the compressibility, giving a
parasitic capacitance which inevitably scales with
C' = C/λ3. The related R⋅C time-constant does not scaledown
with size.
4 SCALING
Based on the component descriptions, the scaling
behaviour of hydraulic systems containing inertances, linear
resistors and capacitors can be analysed. An important
question is if these systems can be made faster by making
them smaller. For this class of systems there are two types
of characteristic time constants: For the filling of a
hydraulic capacitance through a resistive channel the R⋅C
time constants, and for the acceleration of the liquid in a
resistive channel the I / R time constants. For a system S'
that is λ times smaller than a system S in all three
dimensions scaling laws have been derived for two cases.
CONCLUSIONS
The lumped network approach offers a powerful design and
analysis tool for low Re hydraulic systems. We have
illustrated this by rebuilding a common electronic
relaxation oscillator in the hydraulic domain. The hydraulic
astable multivibrator was successfully driven at a pressure
of 104 Pa with ethanol as the medium. The analysis of the
down-scaling of micro-hydraulic systems shows that to
make smaller systems quicker, higher pressures are
required to compensate for the viscous losses. However, a
practical limit is caused by the compressibility of the liquid,
which causes a R⋅C time-constant, which does not scale
down with size